Journal of Babylon University/Pure and Applied Sciences/ No.(5)/ Vol.(21): 2013
Deriving And Calculation Of Point Spread
Function For Array Of Obscured Circular
Synthetic Apertures
Ban Hussein Ali AlrueshdyM , Adnan Falih Hassan Aldehadhawe
University of Kufa/ College of Sciences / Department of Physics
Abstract
This research includes the equation of point spread function (PSF) which has been derived for array of
obscured circular synthetic apertures; it studies the effect of changing the obscuration ratios (ε =0, 0.25, 0.5,
0.75 ), MathCAD programs were used to simulate the equation of the point spread function, we obtain from
the result increasing the intensity of secondary peaks and the best resolving power comparison with
resolving power for obscured individual circular aperture.
‫الخالصة‬
‫تضمن البحث اشتقاق معادلة دالة االنتشار النقطية لصف من الفتحات المركبة الدائرية المعاقة ودراسة تأثير تغير نسب اإلعاقة‬
‫ ووجد من خالل النتائج حصول زيادة في شدة القمم الثانوية وقدرة التحليل‬, ‫) وذلك باستخدام برنامج الماثكاد‬ε=0,0.25,0.5,0.75)
.‫أفضل مقارنتاً مع قدرة تحليل الفتحة الدائرية المنفردة المعاقة‬
Introduction
Most optical systems are used to create images: eyes, cameras, microscopes,
telescopes ,etc. These image forming instruments use lenses or mirrors whose properties,
in terms of geometrical optics(A. Lipson et al.,2011). The image of a point source object
formed by a lens system is known as the point spread function (PSF) of the lens, it is one
of the most complete function for describing the performance of an optical system and
can be extended to include the effects of an obstructed aperture, apodization and any
factor external to the optical system(A.F. Aljebory, 2004). The properties of the optical
system can be described by a point spread function, point spread function is related to the
phase at these exit pupil apertures through a Fourier transform (K.K. Sharma, 2004). A
PSF is an irradiance distribution representing the image of an ideal point source , namely
a PSF is a diffraction limited intensity distribution in response to a point source such as a
distant star (Soon-Jo Chung,2002). Diffraction plays an important role in optical systems
(R. Murugeshan, 2008). An imperfection or distortion in an image is called an aberration,
an aberration can be produced by a flaw in a lens or mirror, but even with a perfect
optical surface some degree of aberration is unavoidable (B. Crowell,2007). The method
adapted to seeing the close objects as separate objects is called resolution. The ability of
an optical instrument to produce distinctly separate images of two objects located very
close to each other is called its resolving power(N. Subrahmanyam et al., 2009; R.
Murugeshan, 2008). Synthetic aperture techniques are commonly used to obtain high
resolution from data acquired using low resolution sensors, these techniques are
commonly used in modern sonar and radar systems, being designated by Synthetic
Aperture Sonar (SAS) and by Synthetic Aperture Radar (SAR) systems, respectively; this
kind of systems is presently used in civil and military applications (P. Marques et
al.,2004). " Synthetic Aperture " Can be defined as a structure to separate optical
systems of large individual aperture function sometimes called " Mosaic" or "Segmented
1819
Mirror", Synthetic aperture is an image system for independent optical system which are
together sharing the image domain(A.F. Aljebory, 2004).
Deriving the Equation of Point Spread Function for Array of Obscured
Circular Synthetic Apertures
The representation of the array of circular synthetic apertures with radius (R) inside it
array of obscured circular synthetic apertures with radius (ɛ R)as shown in Figure (1)
(+1)-(+ɛ)
(
1
N
)(

N
)
(
R
εR
1
N
 y2 )  (
2
N
 y2 )
( 1 y 2 ) 
(  2  y2 )
.
(
1
N
 y 2 )  (
(
2
N
 y2 )
1
N
)  (

N
)
(-1)-(-ɛ)
(b)Obscured Synthetic Aperture N=4,
when N is the Number of Obscured Aperture
(a) Obscured Individual Circular Aperture
Figure (1) The Integral Boundary for Array of Obscured Circular Synthetic Apertures
The ratio between the radius of obscured circular synthetic apertures and the radius of
circular synthetic apertures is called obscuration ratio and it is symbolized (ɛ) (G.C. Sloon
et al., 2003; G.S. Alshabaan,2001; I.M. Alkhafaji,2011)
The equation for array of circular synthetic apertures is given by the relation (see
Figure(1)):
R
x2  y 2  (
) 2 ..…………………….………………….……(1)
N
and when R  1 , then:
x 
1
 y2
N
and
y 
1
N
……………………………(2)
The equation for array of obscured circular synthetic apertures is given by the relation
R 2 ………………………………………………….. (3)
x 2  y 2  (
)
N
x  
2
N
 y 2
and
y  

………………………………(4)
N
The intensity in point spread function for array of obscured circular synthetic apertures,
can be calculated by the relation (G.C. Sloon et al., 2003).
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Journal of Babylon University/Pure and Applied Sciences/ No.(5)/ Vol.(21): 2013
PSF  F
2
 F1  F2
2
………………………………………………..(5)
where
F1 : is the complex amplitude function for array of circular apertures.
F2 : is the complex amplitude function for array of obscured circular apertures.
We can express the complex amplitude in the point (u , v ) in the image plane by
using Fourier transform to pupil function( B.J. Davis et al.,2004; G.A. Cirino et al.,2006;
J.W. Goodman,1998; K.J. Gasvik,2002; S. Wang et al.,2008).
f ( x, y)   ( x, y)e ikw( x. y) …………………………………………………..(6)
where
 ( x , y ) : represents the real amplitude distribution in exit pupil and which is called "
pupil transparency " or " transmission function " and often chooses equal one unit .
e ikw( x, y ) : wave front of aberration function.
w( x , y ) : aberration factor
( x , y ) : exit pupil coordinates (G. Chartier, 2005; M. Bass et al.,2010).
We can express point spread function for circular aperture in its integral form, by
putting the exit pupil coordinates of form ( x , y ) instead of ( x , y ) , for simplicity .
F1 (u , v)  n. f

f ( x, y ) e
i 2 ( ux  vy )
dxdy
…………………………..(7)
y x
and for N from obscured circular apertures (Figure(2)), we get:
 x  x   x j y  y  y j …………………………………………………….…(8)
y
y
yj
x
x
0
xj
Figure (2) Multiple – Obscured Aperture Configuration
1821
N
F1 (u , v)  n. f    f ( x, y )e
i 2 [ u  ( x x j )  v ( y y j )]
dxdy …………………...……..(9)
j 1 y x
N
F1 (u , v)  n. f    f ( x, y )e
i 2 ( u x vy )
.e
i 2 ( u x j  vy j )
dxdy ………………………(10)
j 1 y x
F1 (u , v )  n. f

f ( x, y )e
i 2 ( u x  vy )
N
dxdy. e
i 2 ( u x j  vy j )
………………………..(11)
j 1
y x
By substituting the equation (6) in equation (11), we obtain:
F1 (u , v )  n. f    ( x, y )e
ikw( x , y )
N
.e i 2 ( uxvy) dxdy. e
……………..…(12)
j 1
y x
Let
i 2 ( u x j  vy j )
 ( x, y )  1
F1 (u , v )  n. f   e
ikw( x , y )
N
.e i 2 ( ux vy) dxdy. e
i 2 ( u x j  vy j )
…..………………..(13)
j 1
y x
where k  2

F1 (u , v)  n. f   e
i 2 [ w ( x , y )  ( u x vy )]
N
dxdy. e
i 2 ( u x j  vy j )
…………………….(14)
j 1
y x
By substituting the integral boundary from equation (2) the area of synthetic aperture
includes
F1 (u , v )  n. f

where
1
 y2
N
i 2 [ w ( x , y )  ( u x vy  )]
1
N

1
N
e

N
dxdy. e
i 2 ( u x j  vy j )
…………………(15)
j 1
1
 y2
N
e i  cos   i sin 
 1
 N
F1 (u , v )  n. f  
  1N



 cos2 w( x, y)  u x  vy   i sin 2 w( x, y)  u x  vy dxdy
1

 y2
N

1
 y2
N

N

. cos2 (u x j  v y j ) i sin 2 (u x j  v y j )..............................................................(16)
 j 1

Let
z   2u and
m  2v 
become :
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Journal of Babylon University/Pure and Applied Sciences/ No.(5)/ Vol.(21): 2013
 1
 N
F1 ( z , m)  n. f  
  1N



 cos2w( x, y )  z x   my   i sin 2w( x, y)  z x   my dxdy
1

 y2
N

1
 y2
N

 N

. cosz x j  my j  i sin z x j  my j .........................................................................(17)
 j 1

The intensity distribution on the two axes ( z , m ) is symmetric, so; we can reduce it
to one axis only; let ( m  0) , then equation (17) will take a new form .
 1
 N
F1 ( z )  n. f  
  1N



 cos2w( x, y )  z x  i sin 2w( x, y )  z x dxdy
1

 y2
N

1
 y2
N

 N
. cos( z x j )  i sin( z x j )
 j 1

.....................................................................................(18)

Equation (18) represents the complex amplitude for array of circular synthetic apertures .
The complex amplitude function for array of obscured circular apertures from
equation (6) can be found as:
F2 (u , v)  n. f   f ( x , y )e
i 2 ( u x vy )
dxdy 
………………………………….(19)
y x
By using the same steps and substituting the integral boundary from the equation (4) we
obtain .
 
 N
F2 ( z )  n. f  
 
 N


2 cos2w( x, y )  z x  i sin 2w( x, y )  z xdxdy 

 y2
N

2
N

 y2
N

. cos( z x j )  i sin( z x j ) ..........................................................................................(20)
 j 1



By using the physical conception of the equation (5), we can obtain the following:
 1
 N
 
 1
 N
1
 y2
N
cos2w( x, y )  z x   i sin 2w( x, y )  z x dxdy
 . cos( z x
N
1
 y2
N

PSF  n. f
 N
  

  N
2
N
 y2
j 1
 . cos( z x
2
N
 y2





cos2w( x , y )  z x   i sin 2w( x , y )  z x dx dy 
N


j )  i sin( z x j )
2
j 1
From the relation x  iy
2

j )  i sin( z x j )

 x 2  y 2 the equation above become:
1823



.............(21)

N
 cos2w( x, y )  z xdxdy. e

1
N
2




N


1
N

N
N
sin 2w( x, y )  z xdxdy. e

y
1
N
.....................( 22)
iz x j
j 1
2
 y2
N
N
 sin 2w( x, y )  z xdxdy . e


N

2
N
iz x j
j 1
 y2
 y2
2
N

2
1
N


N
 cos2w( x, y )  z xdxdy . e

1
N

 y2
N
N
iz x j
j 1
1
 y2
N


PSF  n. f
2
1
 y2
N
1
N
iz x j
j 1
 y2
This equation represents the point spread function for array of obscured circular
synthetic apertures.
By substituting the value of normalizing factor (
1
(   2 ) 2
)(B.H. Alrueshd ,2011).in
equation (21),
we obtain:
1
 y2
N
1
N


1
N

PSF 
1

(    )
2
2
N
N
N

2
1
N



1
N
 y2
2
 y2

2
N
izx j
j 1
...................( 23)
N
sin( z x) dxdy. e
izx j
j 1
N
 sin( z x)dxdy. e

N
 y2
N
N
 
N
 y2
N



2
1
N

1
N
 y2
 cos( z x)dxdy. e


izx j
j 1
1
 y2
N


N
 cos( z x)dxdy. e
izx j
j 1
 y2
Equation (23) represents the point spread function for array of obscured circular
synthetic apertures with diffraction – limited system .
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Journal of Babylon University/Pure and Applied Sciences/ No.(5)/ Vol.(21): 2013
1.2
1.2
N=1
N=5
N=6
1
1
0.8
PSF
0.8
PSF
N=1
N=5
N=6
0.6
0.6
0.4
0.4
0.2
0.2
0
1
2
3
4
5
6
7
8
9
0
10
1
2
3
4
6
7
8
9
10
Z
Z
Figure(2)Point Spread Function When (w20=0,R=1, ε=0.25)
Figure(1)Point Spread Function When (w20=0, R=1, ε=0)
1.2
1.2
N=1
N=5
N=6
1
N=1
N=5
N=6
1
0.8
PSF
0.8
PSF
5
0.6
0.6
0.4
0.4
0.2
0.2
0
1
2
3
4
5
6
7
8
9
0
10
2
3
4
5
6
7
8
9
10
Z
Z
Figure(3)Point Spread Function When (w20=0, R=1, ε=0.5)
1
Figure(4)Point Spread Function When (w20=0, R=1, ε=0.75)
1825
Discussion
Through out noticing the results that we have obtained from the MathCAD program for
the point spread function(PSF) for array of obscured circular synthetic apertures, which
has the circular synthetic apertures (N=1,5 and 6) with different obscuration ratios
(ɛ=0,0.25,0.5 and 0.75), for diffraction limited system and the separating distances (R=1)
(where R represents distance from the center of synthetic aperture to the center of
Cartesian coordinates), we can conclude that : By using the obscured circular apertures
system the one part transfers from central spot energy (Airy disk) to secondary peaks in
diffraction pattern. The number of secondary peaks increase with increasing the
obscuration ratio (ɛ), and the resolving power for obscured aperture with obscuration
ratio ɛ=0.75, is better than resolving power with obscuration ratio ɛ=0.5 and ɛ=0.25. This
is shown in Figures(1),(2),(3) and(4), which represent the values of PSF for different
values of obscuration ratios (ɛ=0,0.25,0.5 and 0.75) for diffraction limited system having
individual circular aperture. We notice from the Figures above, with increasing the
obscuration ratio increasing the secondary peaks with clear shape, because increase the
diffraction ratio of the light rays, and increment of deviation of the light passing through
it and appearance the another secondary peaks as result. Figures (1),(2),(3)and (4)
represent the intensity distribution curves of PSF for diffraction limited system with
obscuration ratios (ε=0,0.25,0.5 and 0.75), when the number of apertures (N=5 and 6).
We notice that by increasing the number of apertures we see the positions of secondary
peaks were moved inside (near the PSF axis), This means increasing the intensity of
secondary peaks. We notice from the Figures above an increase in the resolving power
with increasing the obscuration ratio which is equivalent with increment for the number
of apertures which was became best from the obscured individual aperture.
The cause of increase in resolving power when we use a design having the obscured
circular synthetic apertures is the presence of obscuration in circular synthetic apertures
which in turn causes the diffraction of the light passing through it and when increasing
obscuration ratio. The diffraction will increase which means redistribution of the intensity
in the image plane.
The intensity for PSF in this research (array of obscured circular synthetic apertures)
represents the intensity distribution of an array of similarly oriented identical circular
aperture equal intensity distribution of an individual aperture function multiplied by
intensity sum result from a set of point sources arrayed in the same configuration.
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Journal of Babylon University/Pure and Applied Sciences/ No.(5)/ Vol.(21): 2013
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